IMPROVING GEOIDAL HEIGHT ESTIMATES FROM GLOBAL GEOPOTENTIAL MODEL USING REGRESSION MODEL AND GPS DATA

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1 Bulelln Geoinformasi. Jld 1, No, 3, ms, , Disember 1997 Penerbitan Akademik Fakulti Kejuruteraan & Sains Geolnformasi IMPROVING GEOIDAL HEIGHT ESTIMATES FROM GLOBAL GEOPOTENTIAL MODEL USING REGRESSION MODEL AND GPS DATA Khalrul A. Abdullah, Ph.D Faculty ofgeoinformation Science And Engineering, Universiti Teknologi Malaysia, Locked Bag No.791, Jollor Bahru, Johor, Malaysia Abstract COTlvenlionaJly. for most application, position of a point is often referred to the geoid as the reference surface. Thus there is an important need for the knowledge of the geoid undulation in the area where positioning tasks is perfonned, This requirement is made more apparent with the advent of high precision using GPS where the resulting ellipsoid height must be converted to orthometric height. An ideal solution is to use a precise gravimetric solution where the geoidal height at each GPS point is computed and applied. Unfortunately, at the moment there is no such solution available in Malaysia. However. efforts are currently being made to develop a precise gravimetric geoid, For the time being, an altemative method would have to be use and the global geopotential model is one of them. [n order to increase the accuracy of computed geoid height from the geopotentia! model. a regression model is used in conjunction with the GPS data. The resulting accuracy estimates of the geoid height determination increases from around 60 cm 'to about 10 cm leve INTRODUCTION The advent of high precision relative positioning by use of the Global Positioning System (GPS) has opened up an alternative to the classical method of height detennination. Using GPS, it is now possible to detennine routinely, very accurate three-dimensional Cartesian coordinate differences (or baseline vectors) between observation points. This three-dimensional capability therefore provides the possibility of using GPS for height detennination as well as for (horizontal) positioning. Conventionally, topographic maps, engineering design and construction project plans, usually depict relief by means of orthometric height, which differs from the GPS derived ell ipsoidal height. Thus, the application ofgps will be further extended if accurate transformations between GPS eliipso idal height and the orthometric height can be realized. This can be accomplished on the condition that we know the geoid height, which relates the orthometric height to the GPS ellipsoidal height and this requires the knowledge of the geoid. The geoid infonnation can be obtained by the use of one of the following geoid solutions, namely; using a plane surface model, a gravimetric geoid model, and a global geoid model. For small area of less than 10 km, which is not too hilly and the gravity fi,e:ld is smooth, a plane surface can be used to approximate the geoid using control points with known orthometric heights (Hajela, 1990; and Khairul, 1993). For a more accurate estimates, a precise gravimetric solution is required but unfortunately it is still not yet available in Malaysia and work is currently undertaken to rectify this situation, An alternative approach would be to use a global geopotential model to estimate the geoidal height. Khairul (1993) have shown that the accuracy of the height detennined by using a global geoid model in Malaysia is about 0.5 to I.Om. In order to improve the height estimates, a regression model 112

2 Improving GeoId..1Height Eltim..tes From GIob3I Geopotential Model employing four parameters were tested using a GPS network at UTM. This paper anempts to report on the results obtained from the study. 2.0 ORTHOMETRIC HEIGHT ESTIMATION 2.1 GPS Helghtlng The position of points derived from GPS measurements are usually computed in a three..<jimensional Cartesian coordinate system. The basic results of the precise differential GPS survey of a baseline are the Cartesian coordinate differences AX, IiY and liz. Baselines connecting the observed GPS points arc then put through a network adjustment such as LJD-HEIGHT (Khairul," 1994) or GEOLAB (Bitwise. 1991). The resulting X. Y. and Z coordinates of the GPS points are then transfonned, using a reference ellipsoid. into geodetic coordinates in terms of latitude (+). longitude p..) and ellipsoidal heig.ht (h). The following relation relates the orthometric height (H) to the ellipsoidal height (h) : 0<. [ I J where. H - hops NMOOEL [2J hops is the GPS derived ellipsoidal height. and. NMOOEl is the geoidal height derived from a geoid model. or by the relative approach. the orthometric height difference between two GPS points may be deduced from: IiH -lihgps -linmodel [3] From the expressions above, the errors in H in eqn. 12) will depend upon the accuracy of the parameters used in its evaluation. It is generally known that. the differences in h between two points measured simultaneously by GPS are much more precise than h at either of the points. This is because of the presence of systematic errors which, being significantly the same at the two points. cancels in the difference. Similarly, IiNMODEL' is much more precise than the geoid height at either points. This means that for determination requiring highest precision. the approach implied in eqn. (3J is preferred to that in eqn. (2]. 2.2 Derivation of Geoldal Height One way to derive the geoidal height is by employing a global geoid solution. Global geoid solutions are obtained from global geopotential models. which are given as a set of coefficients consisting of a series of spherical harmonic functions. The coefficients of the various terms in the series are determined using a combination of satellite orbit analyses (for the long wavelength geoid features). terrestrial gravity (medium to short wavelength features) and geoid heights measured by satellite altimetry over the ocean (medium to short wavelength features). The geoidal height from a global geoid model NOM is computed from a set of normalized geopotential coefficients using the following equalion: GM '~a)' +- [-. _ 1_ ( ) N GM =-- - L C_cosm..i+S_sinm..i P_ sin; (4J ry..2 r.-0

3 Improving Geoidal Height Estimates From Global Geopotential Model where, n MAX is the maximum degree at which the coefficients are known. -* C nm are the C nm less the zonal coefficients ofthe the normal potential ofthe selected reference ellipsoid. G is the gravitational constant. M is the mass of the earth, including the atmosphere. a is the earth's equatorial radius. r is the distance from the earth's center of mass. ~, ill. are the geocentric latitude and longitude. Pom (sin <l» is the normalized asociated Legendre function y is the nonnal gravity 0, m is the degree and order respectively Generally, the more coefficients there are in a model, the more detailed the model usually is since it contains shorter wavelength information of the earth's gravity field. This means that in general, the best solution to use is one that has been determined ~p to the maximum degree and order of 360, which, theoretically at least, can model features in the geoid with half wavelength ofg.5 degrees or 55 km. In this study, the global geopotential model adopted is the OSU91 A which was developed using 30' by 30' mean gravity anomalies derived from terrestrial and altimetric data (Rapp et al., 1991). These data are then combined with GEM-T2 (Marsh et al., 1989) to produce the model complete to degree and order of 360. This model was chosen on the basis that it is the most up-ta-date global geopotential model made available. 2.3 Improving the Geoid Height Estimates The geoidal height computed using the solution described above may contain biases due to several factors, such as the problem arising from the differences in the GPS and geoid model datums. This is especially apparent in the case of using a global solution where the biases may consist of longwavelength errors contributed by geopotential model errors, bad gravity coverage and a bad elevation datum for the gravity observations, since barometric levelling is commonly used. These biases can be reduced or absorbed by implementing some kind of transfonnation procedure such as that used by Forsberg et al. (1990). The geoid change (N' - NMODEL) due to these biases can be expressed in geodetic coordinates in the fonn of a regression formula (ibid.): [5] By using at least four known geoidal heights, N', in the above equation, the four coefficients in the regression model can be computed. These coefficients are then used in computing the 'correction' that will be applied to NMODEL in deriving the geoid height and the orthometric heights at the other points. 114

4 Improving Geoidal Height Estimates From Global Geopotentlal Model 3.0 THE EXPERIMENT 3.1 The GPS/Helght Network In order to test and evaluate the proposed method of height determination, a network of points with known heights is clearly needed. A network of 10 points with known heights was established within UTM campus for this purpose. The heights of the 10 points are derived using the conventional levelling method. Figure [1.0] shows the distribution of all the to points. The GPS observations were made using three Ashtech""" and one Topcon T '" receivers. A total of24 baselines were processed using the GPPSTIol post-processing software. 3.2 The Tests Using Global Geopotential Model Global geopotential model as discussed previously can be used to derive the geoidal height to correct the ellipsoidal height to give us the orthometric height. The geoidal height is computed using eqn.[4.01 using a set of coefficients. As discussed previously there are quite a number of global geopotentiaj model available that can be utilized but for this study OSU9IA(Rapp et al.) coefficients were used. Table[ 1.0) shows the orthometric heights derived using the OSU91 A coefficients. The r.m.s computed from this model is about 82 em and this accuracy is not suitable for most engineering application. A strategy to improve the orthometric height estimation using the global geopotential model was anempted. This is because the geoidal height computed using the solution described above may contain biases due to several factors, such as the problem arising from the differences in the GPS and geoid model datums. These biases can be reduced or absorbed by implementing some kind of transformation procedure such as that used by Forsberg et aj. (1990) as described in eqn.[5.0). By using at least four known geoidal heights, N', the four coefficients in the regression model can be computed. These coefficients are then used in computing the 'correction' that will be applied to NMODEL in deriving the geoid height and the orthometric heights at the other points. Table[2.0} shows the orthometric heights of GPS points computed in this manner. The resulting r.m.s of 10.4 Cm signifies a significant improvement in the height estimation. GEODETIC COORDINATE TRUE ESTD. DlFF. STATION (WGS 84) ORTH. HEIGHT HEIGHT ~ A h (m) (m) (m) (m) BM BM c BM BM BM BM )' :808 RMS 81.8 cm Table 1.0 Solution using global geopotential model (OSU91A) only 115

5 Improving Geoldal Height Estimates From Global Geopotential Model GEODETIC COORDINATE TRUE ESTD. DlFF. STATION (WGS 84) ORTH. HEIGHT HEIGHT $ A h (m) (m) (m) BM BM BM BM BM BM RMS= 10.4 cm Table 2.0 Solution using OSU9l A and REGRESSION MODEL 4.0 CONCLUSIONS Based on the tests described above, the following conclusions can be made: Using geoida1 height from a global geopotential model may give an accuracy of about 80-cm to I metre to the height detennination. This level of accuracy is below the requirement of many engineering applications. Employing a linear regression to overcome biases that may arise from using a global geopotential model does contribute a significant improvement on the height estimation. For some engineering applications, the use of GPS data in conjunction with additional known height of several points has the potential of replacing the conventional spirit levelling for height detennination

6 Khalrul Anuar Abdullah. Ph.D Be 10 BM 02 Figure 1.0 The Test GPS Network Acknowledgements Assistance rtteived from a number of people in implementing this study is greatly appreciated. Dr. R. H. Rapp for providing {he OSU91 A geopotential coefficients: and the Research and Development Unit,. Universiti Teknologi Malaysia for funding this work. 117

7 Improving Geoidal Height Estimates From Global Geopolenllal Model Usmg Regression Model And GPS Data REFERENCES BiHvise (1991 ). (,tol ah Manual II it \I' i,~ Idcas III, CUlaJa rorsbl'il:. It lin,) F. ~lutlsi II(19 )O). 1I1~1I-1'.-eci,illll (jeilld Ilclg.tll' FIll (;I'S I.cvdling "/"{J(eed,ngl Second!nlanationul Srlllpo~'illlll o{pn.!('i."f! POSIflOJlIIIg h,lth the (dohal I'oJ;IIOIIIII).!. \")'\"fc:m. (Hltl\...'a l'~lll;uja HllJcla. U. (1')')11). Ohl""I"':! (ellljlllet,",-i',-cci",i"" I kil;hl' 11) (d'" ()I1"""II"I" {)'~f SIlI,,11 All'", (if's World New kr,t}. )anum) /h:hruar\' Issue pp:' ~ - ~ l) I..:hairul.\. Abdullah (1')')J). Urr/wllle"""' 1Ii!1~1i/ emma/lui? J-rom A ((liiiiji/i(/11i1i/ of (ii'.\'./ild l{'i'r~,'lri{j1 (Ieodttl'"!)(Jta Ph D Tile, " Dc pari lllcll I "I ';"n'", illg. I!Iliv".-sil\' 01" N",I'C<1,tk ljlwfl '1 yne. lj K h:hhirul.\.,\btlullah (I')')") )-111 I(illl S)'mposl/i/ll 0" De!o'-JIIallO!I>. b"1i1hlll rllr"", :\Iarsh. S. '\ 1.. F..1. t.ercll. Il. II. l'ullll y. S. 1\ :10,1«(>. L C. l'a"ii,. G. B. I'ald. 1>. ~. ('hin n. and C. A. Wag,ner ( 1988). A.n impro\'ed ","del 01 Ihe b'rth, (if<l\ Ilali"Jldl I'kld "GLM I 1* iv,isil!ec!r Memo II)O--Ili Rapp. R.H.. \'.~1. Wang. am] N.";. l'a\'lis (1991). TIl<; Ohio StJlC I 991 G~"pul~nlial and Sea Surfacc Topography Harmonic Coeff'Clcnl Mode]', Reporl.\'O -III! oflhl! f)l!pl o.!g~odl!lic.~cle/lce. I"h~ Ohio Slale ljnlvcr,ily. Columhus. Ohiu. 91 pp" 118